We have a \(T\mathbf {C} ^*\times \mathbf {C} ^*\)  torus action on \({F_{\textnormal {\textsc {mb}}}}\) , given by translation on the smooth locus, and which extends to an action on all of \({\widehat{F}_{\textnormal {\textsc {mb}}}}\) . This gives us an action on its coherent sheaves \(\operatorname{Coh}({\widehat{F}_{\textnormal {\textsc {mb}}}})\)  and thus on the moduli space \(M_{\beta }^{{\widehat{F}_{\textnormal {\textsc {mb}}}}}\) . This action preserves the canonical class and is compatible with the symmetric obstruction theory. We can use the motivic nature of the Behrend function weighted Euler characteristic to stratify the moduli space under this group action [1]}, [2]}. The nontrivial torus orbits make no contribution to \(e(M_{\beta }^{{\widehat{F}_{\textnormal {\textsc {mb}}}}},\nu )\) , and we can reduce to considering only the \(T\) -fixed points of the moduli space \((M_{\beta }^{{\widehat{F}_{\textnormal {\textsc {mb}}}}})^T\) .
